Special series sums

Let $\alpha,\ \alpha+2$, $\alpha\in\mathbb Z$, be the roots of the quadratic equation
$$x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots+(x+n-1)(x+n+1)=4n$$
for some $n\in\mathbb N$. Then $n+\alpha$ is equal to :

If $\displaystyle\sum_{k=1}^na_k=6n^3$, then $\displaystyle\sum_{k=1}^6\left(\frac{a_{k+1}-a_k}{36}\right)^2$ is equal to ________.

The sum $\dfrac{1^3}1+\dfrac{1^3+2^3}{1+3}+\dfrac{1^3+2^3+3^3}{1+3+5}+\cdots$ up to 8 terms, is :

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

Let for some $\alpha\in\mathbb R$, $f:\mathbb R\to\mathbb R$ be a function satisfying $f(x+y)=f(x)+2y^2+y+\alpha xy$ for all $x,y\in\mathbb R$. If $f(0)=-1$ and $f(1)=2$, then the value of $\displaystyle\sum_{n=1}^5\left(\alpha+f(n)\right)$ is:

Let the sum of the first $n$ terms of an A.P. be $3n^2+5n$. Then the sum of squares of the first 10 terms of the A.P. is:

$\displaystyle\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:

If $f:\mathbf N\to\mathbf Z$ is defined by
$$f(n)=\begin{vmatrix}n&-1&-5\\-2n^2&3(2k+1)&2k+1\\-3n^3&3k(2k+1)&3k(k+2)+1\end{vmatrix},\quad k\in\mathbf N,$$
and $\displaystyle\sum_{n=1}^{k}f(n)=98$, then k is equal to :

If the sum of the first 10 terms of the series $\dfrac1{1+1^4\times4}+\dfrac2{1+2^4\times4}+\dfrac3{1+3^4\times4}+\dfrac4{1+4^4\times4}+\dots$ is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

The value of $1^3-2^3+3^3-\dots+15^3$ is:

The sum $1+\dfrac12\left(1^2+2^2\right)+\dfrac13\left(1^2+2^2+3^2\right)+\dots$ upto 10 terms is equal to :

Let

$$\alpha = \frac{1}{\sin60^\circ\sin61^\circ} + \frac{1}{\sin62^\circ\sin63^\circ} + \cdots + \frac{1}{\sin118^\circ\sin119^\circ}.$$

Then the value of

$$\left(\frac{\operatorname{cosec}1^\circ}{\alpha}\right)^2$$

is _____.

Let $7\overbrace{5\cdots5}^{r}7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7\overbrace{5\cdots5}^{98}7$. If $S = \dfrac{7\overbrace{5\cdots5}^{99}7 + m}{n}$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is _____.

Let $a_1, a_2, a_3, \ldots$ be an arithmetic progression with $a_1 = 7$ and common difference 8. Let $T_1, T_2, T_3, \ldots$ be such that $T_1 = 3$ and $T_{n+1} - T_n = a_n$ for $n \ge 1$. Then, which of the following is/are TRUE?

For a natural number n, let $\alpha_n=19^n-12^n$. Then, the value of $\dfrac{31\alpha_9-\alpha_{10}}{57\alpha_8}$ is ______.

Let $A=\displaystyle\sum_{i=1}^{10}\sum_{j=1}^{10}\min\{i,j\}$ and $B=\displaystyle\sum_{i=1}^{10}\sum_{j=1}^{10}\max\{i,j\}$. Then $A+B$ is equal to ______.

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\dfrac{2e^{2x}}{e^{2x}+e}$. Then $f\left(\dfrac1{100}\right)+f\left(\dfrac2{100}\right)+f\left(\dfrac3{100}\right)+\dots+f\left(\dfrac{99}{100}\right)$ is equal to ______.

If the sum of the first ten terms of the series
$$\frac15+\frac2{65}+\frac3{325}+\frac4{1025}+\frac5{2501}+\dots$$
is $\dfrac mn$, where m and n are co-prime numbers, then $m+n$ is equal to ______.

Let $S=2+\dfrac67+\dfrac{12}{7^2}+\dfrac{20}{7^3}+\dfrac{30}{7^4}+\dots$. Then $4S$ is equal to

The integral $\displaystyle\int_0^1\frac1{7^{\left[\frac1x\right]}}dx$, where $[\cdot]$ denotes the greatest integer function, is equal to